Weak ergodicity breaking in an anomalous diffusion process of mixed origins

Abstract

The ergodicity breaking parameter is a measure for the heterogeneity among different trajectories of one ensemble. In this report this parameter is calculated for fractional Brownian motion with a random change of time scale, often called "subordination". We proceed to show that this quantity is the same as the known CTRW case.PACS numbers: 05.40.Fb,05.40.Fb "Weak ergodicity breaking" is a term which occurred in the spotlight a few years ago, see for instance [1][2][3][4][5][6] with respect to observables showing aging behavior. The term "aging" applies to the specific behavior pertinent to the values of an observable measured at two distinct instants of time, like the mean squared displacement (MSD) in the time interval between t 1 and t 2 , ∆X 2 (t 1 , t 2 ) as depending on the age t 1 of the system (assumed to be prepared at t = 0) at the beginning of observation. Aging here essentially refers to a non-stationarity of the observed process, in which ∆X 2 (t 1 , t 2 ) cannot be expressed via the time lag τ = t 2 − t 1 only and keeps a considerable dependence on t 1 as a stand-alone variable.Let us concentrate now on the ergodicity properties the squared displacement during the time interval τ in a measurement starting at t. In the case when the data acquisition in each single run of the process takes time from t = 0 to t = T , the time average of ∆X 2 (t, t + τ ) can be performed:The process is considered ergodic provided ∆X 2 (τ ) = ∆X 2 (t, t + τ ) . The discussion of the ergodicity implies that the corresponding limit does exist in some probabilistic sense (say, in probability) and is equal to the ensemble mean. Note that the time-average over the data acquisition interval removes the explicit t-dependence. For non-stationary processes the ensemble mean at any t depends on t, while the time-integration over the data acquisition interval removes this dependence. Therefore, even provided the integral above converges in a whatever sense, it cannot converge to all ∆X 2 (t, t + τ ) simultaneously, and our process is trivially non-ergodic.Whether the random process is stationary or not, one can ask how diverse are its different realizations, i.e. how different are two trajectories of the process with respect to some time-averaged observable O(τ ; t) (in our case the MSD, O(τ ; t) = ∆X 2 (t, t + τ )). To this purpose one considers a fixed-T approximation to eq.(1)At any finite T the value of O(τ ) T is a random variable. For both, stationary and non-stationary processes O(τ, t), one can consider a parameter describing the strength of fluctuations of O(τ ) T . As a measure of homogeneity or heterogeneity of different trajectories one can take the relative dispersion of the O(τ ) T in different realizations of the process:The parameter J (often called "ergodicity breaking parameter") shows how different are different trajectories of the process with respect to the observable O. For stationary processes, vanishing of J in the long time limit indeed implies ergodicity [7], and its non-vanishing witnesses against erg...

Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes

“…Continuous time random walk processes with scale free waiting time distribution have a finite value for EB even in the limit ∆/T = 0 [26], similar to HDPs [33][34][35][36], while for scaled Brownian motion the ergodicity breaking parameter approaches zero in this limit [45,46].…”

“…As examples we mention continuous time random walk processes with scale free distributions of waiting times [7,[25][26][27][28][29]31], correlated continuous time random walks [44], as well as diffusion processes with space [32][33][34][35][36] and time [32,38,45,46] dependent diffusion coefficients and their combinations [47]. We also mention ultraslow diffusion processes with a logarithmic form for x 2 (t) and linear lag time dependence (8) of the time averaged MSD [48] as well as the ultraweak ergodicity breaking of superdiffusive Lévy walks [49].…”

Wiener-Khinchin Theorem for Nonstationary Scale-Invariant Processes

“…SBM is maybe the simplest way to construct a stochastic process that exhibits anomalous diffusion. It is defined as x SBM (t) = x(t α ), where x(t) is a Brownian motion [22,23]. Its autocorrelation function reads for τ > 0,…”

Langevin formulation of a subdiffusive continuous-time random walk in physical time

“…However, differently from Brownian motion, it is strongly nonstationary [30]. Furthermore, Y * (t) turns out to be the mean-field approximation of the CTRW, as it describes the motion of a cloud of random walkers performing CTRW motion in the limit of a large number of walkers [29]. Recent investigation has also shown that SBM exhibits rich aging properties, which strongly differentiates it from the standard BM [42].…”

“…(1), while still showing distinct features if we look at other properties, like the multipoint correlation functions [22][23][24][25]. Among the most commonly applied to data analysis, we find the continuous-time random walk (CTRW) [2,26] and the scaled Brownian motion (SBM) [27][28][29][30].…”